# Guide to Non-linear Dynamics in Accelerator Physics/Guided Bibliography

## E. Forest, "Beam Dynamics- A New Attitude and Framework"Edit

Very good source of informations. The topics are somehow mixed and sometime it is difficult to follow linearly. Here there is a selection of relevant sections order by topic:

- tracking: from em field to one turn map around a fixed point
- 3.2.2 connecting pieces
- 3.3.3 misaligments
- 8.1.4 fixed point search
- 10 misaligments (includes bendings)
- 11.3 yoshida integrators
- 11.3.2.2 s dependent first order integrators (maybe kick map)
- 12.1 Hamiltonian splittings for magnets

- linear one turn map:
- 2.4.1 2.4.2 basics of "diagonalization"
- 4.5 linear motions
- 4.5.1 symplectic
- 4.5.2 dissipative

- linear lattice functions: block matrix reduction
- 2.5 betaoids and tune
- 4.4.1 betaoids

- non linear motion: non linear normalization
- 2.3 definitions and properties
- 2.4.3 normal forms algorithm
- 3.4.3 DA equivalence between maps around fixed points and power series
- 4.4 dragt-finn factorization
- 4.4.2 normal forms algorithm
- 5.1.2 terms of the normalized hamiltonians, chromaciticies

- resonances
- 5.1.3 definitions
- 5.2 resonances basis
- 6.3 harmonic driving terms
- 6.4 resonance basis for non symplectic vector fields

- coasting beam
- 7.3.2 coasting beam, etaoids

- analytical perturbation theory
- 8.4 basics
- 9.1 examples (quantum as well)

- damping
- 15 radiation
- pag. 418 interlude (is emittance growth intrinsic or corse graining?)
- 15.3 etaoids betaoid with damping

## L. Michelotti, Intermediate Classical Dynamics with Applications to Beam PhysicsEdit

Includes many of the concepts used in Forest's book. Here they are spelled out a bit more explicitly. L.M. makes a point to generalize phase space to a manifold. (The subject of dynamics is described as "vector fields on a manifold"). This allows for phase space to have more complicated topology than R^n. For example, the phase space of a pendulum is a torus. The book culminates in a description of the Forest, Irwin, Berz normal form algorithm. This is described quite formally, with emphasis on commutative diagrams and solving the homological equations at each step. Although the purpose of such an algorithm (mainly to find amplitude dependent tune shift for a general one turn map) is described clearly at places, it is easy to get lost in all the symbols. Reading chapters 3 and 4 describe the models that one is to keep in mind, and the physics that is to be extracted from the general perturbation theory algorithms of chapters 5 and 6.

The epilogue begins with a quote by Saul Bellow describing the breakdown of language. One can't help but feel that this is commentary on the state of non-linear dynamics in accelerator physics. The break between old methods of Hamiltonian mechanics and new left a descriptive gap in the field. Michelotti has tried valiantly to bridge that gap. Although one may feel that he isn't entirely successful in formulating the symbols and language to describe the physics and the algorithms, this book is the closest to solid ground one may find when trying to navigate some of the literature in this field.

## Guide to papers on non-linear dynamicsEdit

- http://www.slac.stanford.edu/spires/find/hep/www?irn=467014
- Lie Series and Invariant Functions for Analytic Symplectic Maps
- A.J. Dragt , J.M. Finn .
- 1976
- J.Math.Phys.17:2215-2227,1976
- http://link.aip.org/link/?jmp/17/2215
- Dragt-Finn factorization. Good introduction to Lie tools.

- http://www.slac.stanford.edu/spires/find/hep/www?irn=614939
- Method of transfer maps for linear and nonlinear beam elements
- A.J. Dragt .
- 1979
- IEEE Trans.Nucl.Sci.26:3601-3603,1979
- http://accelconf.web.cern.ch/AccelConf/p79/PDF/PAC1979_3601.PDF
- Application of Dragt-Finn factorization to accelerators

- http://www.slac.stanford.edu/spires/find/hep/www?irn=398560
- TRANSFER MAP APPROACH TO THE BEAM-BEAM INTERACTION
- Alex J. Dragt .
- 1979
- Not found
- Not available , asked Fermilab
- Not read

- http://www.slac.stanford.edu/spires/find/hep/www?irn=1014870
- Analysis of the Beam-Beam Interaction Using Transfer Maps. (Talk)
- A.J. Dragt , O.G. Jakubowicz .
- 1980
- Stanford 1980 , Proceedings , Beam-beam Interaction
- Not available
- Not read

- http://www.slac.stanford.edu/spires/find/hep/www?irn=1010344
- Charged Particle Beam Transport Using Lie Algebraic Methods
- D.R. Douglas , A.J. Dragt .
- 1981
- IEEE Trans.Nucl.Sci.28:2522-2524,1981
- http://accelconf.web.cern.ch/AccelConf/p81/PDF/PAC1981_2522.PDF
- Example of factorization for beam lines elements.

- http://www.slac.stanford.edu/spires/find/hep/www?irn=1010476
- Exact Numerical Calculation Of Chromaticity In Small Rings
- A.J. Dragt .
- 1981
- IEEE Trans.Nucl.Sci.28:2627-2629,1981
- http://accelconf.web.cern.ch/AccelConf/p81/PDF/PAC1981_2627.PDF
- Determination of chromaticity from the eigenvalues of the energy dependent
- transfer map. Use of variational calculus is a bit cryptic.

- http://www.slac.stanford.edu/spires/find/hep/www?irn=1224344
- Lectures On Nonlinear Orbit Dynamics.
- A.J. Dragt .
- 1981
- AIP Conf.Proc.87:147-313,1982
- http://link.aip.org/link/?APC/087/147
- General introduction to hamiltonias and lie methods. 167 pages

- http://www.slac.stanford.edu/spires/find/hep/www?irn=1208250
- Lie Algebraic Methods For Particle Tracking Calculations.
- D.R. Douglas , A.J. Dragt .
- 1983
- LBL-16008
- Not available
- Not read

- http://www.slac.stanford.edu/spires/find/hep/www?irn=1276158
- Single Particle Stability
- M. Cornacchia
*et al.*. - 1983
- Ann Arbor 1983, Proceedings, Accelerator Physics Issues For A Superconducting Super Collider
- Not available
- Not read

- http://www.slac.stanford.edu/spires/find/hep/www?irn=1255088
- MARYLIE: The Maryland Lie Algebraic Transport and Tracking Code
- D.R. Douglas , A.J. Dragt .
- 1984
- IEEE Trans.Nucl.Sci.30:2442-2444,1983
- http://accelconf.web.cern.ch/AccelConf/p83/PDF/PAC1983_2442.pdf
- Marylie code, analysis limited to F2, F3